Still no participant

Still no reviews

- To point out that iterative processes and convergence of sequences occur in many areas of mathematics, and to develop a general context in metric spaces.
- To provide a basic course in analysis.
- To reinforce ideas of proof

- Understand the Euclidean distance function on R
and appreciate its properties, and state and use the Triangle and Reverse Triangle Inequalities for the Euclidean distance function on R^{n}^{n} - Explain the definition of continuity for functions from R
to R^{n}and determine whether a given function from R^{m}to R^{n}is continuous^{m} - Explain the geometric meaning of each of the metric space properties and be able to verify whether a given distance function is a metric
- Distinguish between open and closed balls in a metric space and be able to determine them for given metric spaces
- Define convergence for sequences in a metric space and determine whether a given sequence in a metric space converges
- State the definition of continuity of a function between two metric spaces.

Module-I (T-5hr P-0hr P-6hr)

Metric Spaces: Definition and examples, open and closed spheres,

Project-1: Write a report on applications of metric spaces.

Project-2: Show that the class denote a set of all bounded sequence of real numbers is a metric space.

Project-3: Show that the class of all sequences is a metric space.

Module-II (T-3hr P-0hr P-4hr)

Neighbourhoods, Interior points. Open set

Project-4: Write a report on relation between boundary points, limit points. And their applications.

Project-5(Flip class): Prove theorems on Neighbourhoods, Interior points

Module-III (T-5hr P-0hr P-6hr)

Closed set, boundary points, limit points and isolated points, closure of a set, dense sets.

Project-6: Write a report on difference between open set closed set and limit point , interior point with appropriate examples.

Project-7: Write the difference between boundary points, limit points and isolated points explain through examples.

Project-8 (Flip class): Prove theorems on closed set and dense sets.

Module-IV (T-4hr P-0hr P-6hr)

Separable metric spaces, sequences in metric space, convergent sequences, Cauchy sequences

Project-9: Write a report on applications of convergent sequences and Cauchy sequences.

Project-10: Every convergent sequence is bounded. Is the reverse true? Justify your answer with suitable example.

Project-11(Flip class): Prove theorems on convergent sequences and Cauchy sequences.

Module-V (T-4hr P-0hr P-4hr)

Complete metric space, subspace, Cantor’s theorem. Continuous functions,

Project- 12(Flip class): Prove theorems on Complete metric space, subspace.

Project-13(Flip class): : Prove that addition, subtraction, multiplication and division (with condition) of continuous functions are continuous.

Module-VI (T-4hr P-0hr P-6hr)

Uniform continuity, Connectedness, connected subsets of separated sets.

Project-14(Flip class): Prove theorems on Uniform continuity and Connectedness.

Project- 15: Write the difference between continuity and uniform continuity and explain through a proper example..

Project-16 (Flip class): Prove theorems on connected subsets of separated sets.

Module-VII (T-5hr P-0hr P-4hr)

Disconnected sets, contraction mappings, Banach fixed point theorem.

Project- 17(Flip class): Geometrical interpretation of connected and disconnected sets.

Project-18: Write some applications of Banach fixed point theorem.

Text Books:

- K. Jain and K. Ahmad, Metric Spaces, Narosa Publishing House, New Delhi.
- Methods of Real analysis-R.G. Goldberg. Chapters: 4(4.2,and 4.3), 5,and 6 (up to 6.8)

Reference Books:

- S.C. Malik and S. Arora – Mathematical Analysis (New Age International)
- Principles of Mathematical Analysis – Walter Rudin, III Edition, McGRAW-Hill Book Co.

Introduction

PPT link: https://www.slideshare.net/NaliniSPatil/metric-space-151515844

Study Material: Introduction to Metric spaces with examples

youtube link: https://youtu.be/6CLl5xx5X-Y

Examples on Metric Spaces

Study Material: http://more examples

Youtube link: https://youtu.be/etP21xln8iQ

Project-1: Write a report on applications of metric spaces.

Theorems on Metric Spaces.

Study Material: Theorems on metric space

youtube link: https://youtu.be/Af03P1xVNSs?list=PLbMVogVj5nJSxFihV-ec4A3z_FOGPRCo-

Project-3: Show that the class of all sequences is a metric space.

Project-4: Write a report on relation between boundary points, limit points. And their applications

Project-5(Flip class): Prove theorems on Neighbourhoods, Interior points

Project-7: Write the difference between boundary points, limit points and isolated points explain through examples

Project-8 (Flip class): Prove theorems on closed set and dense sets.

sequences in metric space

PPT link: https://www.slideshare.net/VishangSingh/analysis-sequences-and-bounded-sequences

convergent sequences

youtube link: https://youtu.be/DZT8o7Y08Mw

youtube link: https://youtu.be/FgRslqEYTJI

Project-9: Write a report on applications of convergent sequences and Cauchy sequences

Project-11(Flip class): Prove theorems on convergent sequences and Cauchy sequences.

**Project-12(Flip class): Prove theorems on Complete metric space, subspace.**

Continuous functions

PPT link: https://www.slideshare.net/sumanmathews/continuous-functions

youtubelink : https://youtu.be/hXkQqCBLRp8

Project- 13: Prove that addition, subtraction, multiplication and division (with condition) of continuous functions are continuous.

Theorems on Continuous functions

PPT link: https://www.slideshare.net/VishveshJasani/continuity-of-a-function

**Project-14(Flip class): Prove theorems on Uniform continuity and Connectedness. **

Project- 15: Write the difference between continuity and uniform continuity and explain through a proper example

Project-16 (Flip class): Prove theorems on connected subsets of separated sets.

Project- 17(Flip class): Geometrical interpretation of connected and disconnected sets.

Project-18: Write some applications of Banach fixed point theorem.

Kali Prasad Rath received his M.Tech. in Computer Science and & M.Sc. in Mathematics from Berhampur University.This is his 10th year at Centurion University of Technology and Management and his 17th year in teaching profession. He is associated with qualitative study on theory of difference equation.

## Recent Comments